Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Adaptive Experimental Design Using Shrinkage Estimators

Evan T. R. Rosenman, Kristen B. Hunter

TL;DR

The paper tackles adaptive experimental design in multi-armed trials with a control by leveraging Stein-like shrinkage estimators that borrow information across treatment effects. It develops a Gaussian-quadratic risk representation enabling efficient online computation and proposes a greedy adaptive rule to minimize shrinker risk, contrasted with a sequential Neyman approach. Through oracle analyses and extensive simulations, it shows that shrinkage-based adaptivity can substantially reduce estimation error, particularly under low signal-to-noise and sparse effect settings, with allocations tending to place more weight on the control arm. While promising, the approach also raises inference and exploration-exploitation considerations, and points toward future work on probabilistic policies, stopping rules, and covariate-informed shrinkage estimators for causal estimation in adaptive trials.

Abstract

In the setting of multi-armed trials, adaptive designs are a popular way to increase estimation efficiency, identify optimal treatments, or maximize rewards to individuals. Recent work has considered the case of estimating the effects of K active treatments, relative to a control arm, in a sequential trial. Several papers have proposed sequential versions of the classical Neyman allocation scheme to assign treatments as individuals arrive, typically with the goal of using Horvitz-Thompson-style estimators to obtain causal estimates at the end of the trial. However, this approach may be inefficient in that it fails to borrow information across the treatment arms. In this paper, we consider adaptivity when the final causal estimation is obtained using a Stein-like shrinkage estimator for heteroscedastic data. Such an estimator shares information across treatment effect estimates, providing provable reductions in expected squared error loss relative to estimating each causal effect in isolation. Moreover, we show that the expected loss of the shrinkage estimator takes the form of a Gaussian quadratic form, allowing it to be computed efficiently using numerical integration. This result paves the way for sequential adaptivity, allowing treatments to be assigned to minimize the shrinker loss. Through simulations, we demonstrate that this approach can yield meaningful reductions in estimation error. We also characterize how our adaptive algorithm assigns treatments differently than would a sequential Neyman allocation.

Adaptive Experimental Design Using Shrinkage Estimators

TL;DR

The paper tackles adaptive experimental design in multi-armed trials with a control by leveraging Stein-like shrinkage estimators that borrow information across treatment effects. It develops a Gaussian-quadratic risk representation enabling efficient online computation and proposes a greedy adaptive rule to minimize shrinker risk, contrasted with a sequential Neyman approach. Through oracle analyses and extensive simulations, it shows that shrinkage-based adaptivity can substantially reduce estimation error, particularly under low signal-to-noise and sparse effect settings, with allocations tending to place more weight on the control arm. While promising, the approach also raises inference and exploration-exploitation considerations, and points toward future work on probabilistic policies, stopping rules, and covariate-informed shrinkage estimators for causal estimation in adaptive trials.

Abstract

In the setting of multi-armed trials, adaptive designs are a popular way to increase estimation efficiency, identify optimal treatments, or maximize rewards to individuals. Recent work has considered the case of estimating the effects of K active treatments, relative to a control arm, in a sequential trial. Several papers have proposed sequential versions of the classical Neyman allocation scheme to assign treatments as individuals arrive, typically with the goal of using Horvitz-Thompson-style estimators to obtain causal estimates at the end of the trial. However, this approach may be inefficient in that it fails to borrow information across the treatment arms. In this paper, we consider adaptivity when the final causal estimation is obtained using a Stein-like shrinkage estimator for heteroscedastic data. Such an estimator shares information across treatment effect estimates, providing provable reductions in expected squared error loss relative to estimating each causal effect in isolation. Moreover, we show that the expected loss of the shrinkage estimator takes the form of a Gaussian quadratic form, allowing it to be computed efficiently using numerical integration. This result paves the way for sequential adaptivity, allowing treatments to be assigned to minimize the shrinker loss. Through simulations, we demonstrate that this approach can yield meaningful reductions in estimation error. We also characterize how our adaptive algorithm assigns treatments differently than would a sequential Neyman allocation.
Paper Structure (32 sections, 6 theorems, 87 equations, 7 figures, 4 tables)

This paper contains 32 sections, 6 theorems, 87 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. SURE and Candidate Estimators
  4. Notation and Assumptions
  5. Difference-in-Means Estimator
  6. Stein's Unbiased Risk Estimate
  7. Candidate Estimators
  8. Oracle Designs
  9. A Modified Neyman Allocation
  10. Minimizers in the Low Signal-to-Noise Ratio Regime
  11. Bock's Estimator
  12. SURE-Min Estimator
  13. Dimmery's Estimator
  14. Simulated Static Allocations
  15. Computing Risk-Minimizing Allocations
  16. ...and 17 more sections

Key Result

Lemma 1

Suppose $\boldsymbol X \sim \mathcal{N}\left(\boldsymbol \mu, \boldsymbol \Sigma\right) \in \mathbb{R}^K$ for non-degenerate $\boldsymbol \Sigma \in \mathbb{R}^{K \times K}$. We consider an estimator of $\boldsymbol \mu \in \mathbb{R}^K$ of the form for $g(\cdot)$ differentiable and $L_2$ integrable. Define $\mathcal{J}_{g}( \boldsymbol X )$ as the Jacobian matrix of $g(\cdot)$ evaluated at $\bol

Figures (7)

  • Figure 1: Oracle allocations for $K=12$ and $N=1,000$ with low control variance, and $\kappa=0$. All shrinker-risk-minimizing designs shift mass toward the control arm relative to the Neyman allocation, with the largest control shift under Bock's estimator $\boldsymbol {\hat{\delta}_B}$. SURE-min $\boldsymbol {\hat{\delta}_S}$ and Dimmery $\boldsymbol{\hat{\delta}_{\textbf{D}}}$ exhibit the same trend but with smaller magnitude.
  • Figure 2: Oracle allocations for $K=12$ and $N=1,000$ with low control variance, $\kappa=9$, and dense $\boldsymbol{\tau}$. Compared to the $\kappa=0$ case (Figure \ref{['fig:alloc_controlshift_delta0']}), the shrinkage-driven preference for allocating additional units to control is substantially reduced for all estimators, especially the Bock estimator (though it still favors the control arm most among the three shrinkers).
  • Figure 3: Oracle allocations for $K=12$ and $N=1,000$ with low control-variance regime, $\kappa=9$, and sparse $\boldsymbol{\tau}$. Here, we see two key trends. First, the control reallocation for Bock's estimator is much larger than in the otherwise-identical case in which $\boldsymbol \tau$ is dense (Figure \ref{['fig:alloc_controlshift_delta9']}). This reflects the dependency of the minimizing allocation for Bock's estimator on the signal density. Second, we see that the minimizing allocation for Dimmery's estimator -- but not the Bock or SURE-min estimator -- tends to allocate more units to the treatment arm with strong signal when $\boldsymbol \tau$ is sparse.
  • Figure 4: $\kappa=0$.
  • Figure 7: $\boldsymbol \tau$ dense, $\kappa=3$.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Lemma 1: SURE
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • proof
  • proof
  • proof
  • proof
  • ...and 2 more